The US secretary of state, Mike Pompeo, in sunglasses, speaks with his aides after a meeting in Pyongyang with senior North Korean official Kim Yong-chol.
Photograph: Andrew Harnik/AFP/Getty Images

Talks between senior US and North Korean officials in Pyongyang over disarmament and improving bilateral relations entered a second day on Saturday with a testy exchange between the secretary of state, Mike Pompeo and his counterpart, Kim Yong-chol.

Pompeo and a US delegation are in North Korea in an effort to persuade the regime to take concrete steps that Donald Trump claimed he was promised by Kim Jong-un in Singapore on 12 June. He has said the North Korean leader said he would destroy a missile engine testing site and then organise the repatriation of remains of US soldiers killed in the Korean war. Pompeo is also asking for more substantial steps towards disarmament, reportedly including an inventory of the North Korean arsenal of warheads and missiles.

Early on Saturday morning, Pompeo left the Pyongyang guesthouse where the US delegation is staying to make a call through secure communications to Trump at his New Jersey golf club. The national security advisor, John Bolton, and the White House chief of staff, John Kelly, also took part in the call.

Pompeo briefed them on his initial round of talks on Friday with Kim Yong-chol, a former spy chief and vice chair of the ruling party. As well as a working session on Friday, the two men had dinner, which the state department spokeswoman, Heather Nauert, described as “relationship building”. Nauert said that at times they were “cracking jokes” and “exchanging pleasantries.”

She said that progress had been made in Friday’s discussions, adding that the talks had covered the process of denuclearisation and the repatriation of American remains. She added: “we expect them to live up to their commitments.” The two also established working groups to deal with what Nauert described as the “nitty gritty stuff” of outstanding issues between the two countries.

She said Pompeo had been “very firm” in seeking three basic goals: complete denuclearization of North Korea, security assurances, and the repatriation of remains.

When Pompeo returned to the guesthouse after phoning on Saturday, Kim asked Pompeo how he had slept, adding: “we did have very serious discussion on very important matters yesterday. So thinking about those discussions you might have not slept well last night.”

“Director Kim I slept just fine,” Pompeo replied. He went on to point the importance of their task of implementing what Trump and Kim agreed in Singapore and set off on the “the path toward complete denuclearisation”.

“Of course it is important,” Kim said, but added: “There are things that I have to clarify.”

Pompeo replied: “There are things that I have to clarify as well.”

The secretary of state is under time pressure to produce results by August, when the US and South Korea were due to hold joint military exercises. Those exercises were cancelled on Trump’s orders in Singapore as an up-front concession. Adding to the pressure, the president has repeatedly claimed that the testing site has already been destroyed, and that the soldiers’ remains have been sent back, neither of which has happened.

Trump has also made extravagant claims about what was agreed in Singapore. At a rally in Montana on Thursday, he claimed: “We signed a wonderful paper saying they’re going to denuclearise their whole thing. It’s going to all happen.”

In a joint statement with Trump, Kim agreed to move towards “complete denuclearisation” but that has been a stock phrase in North Korean rhetoric since 1992 and signifies a vague and long-term process of multilateral disarmament on the Korean peninsula. Since the Singapore meeting, satellite images and intelligence leaks have suggested that North Korea is upgrading critical parts of its nuclear programme.

“Our leaders made commitments at the Singapore summit on the complete denuclearisation of North Korea and outlined what a transformed US-DPRK relationship could look like,” Pompeo said on his way to Pyongyang.

“Since the summit the consultations have continued. On this trip I’m seeking to fill in some details on those commitments and continue the momentum toward implementation of what the two leaders promised each other and the world. I expect that the DPRK [North Korea] is ready to do the same.”

Even while Pompeo has been in Pyongyang, new satellite pictures were published by the website 38 North, showing work had been completed on the cooling plant for a new nuclear reactor.

In Pyongyang on Friday, Pompeo met Kim Yong-chol, North Korea’s former spy chief and vice-chair of the ruling party, in an extensive guest house in Pyongyang, close to a vast mausoleum where North Korea’s former leaders, Kim Il-sung and Kim Jong-il, are buried.

Welcoming Pompeo and the US team to the first session of talks, Kim Yong-chol, said: “The more you come, more trust we can build between one another.”

“Today’s meeting is a really meaningful meeting,” the 73-year-old general said.

Pompeo replied: “Yes, I agree I look forward to it and I count on it being very productive.”

Noting it was his third visit to Pyongyang, he joked: “If I come one more time, I will have to pay taxes here.”

Mike Pompeo to give Kim Jong-un 'Rocket Man CD' during nuclear talks

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Earlier, Pompeo laughed but did not reply to a question about a South Korean report that he had brought a CD of Elton John’s Rocket Man, as a gift for Kim Jong-un. “Rocket Man” was the nickname Trump gave to Kim when tensions between the two countries reached a peak last year.

The meeting lasted two hours and 45 minutes and Pompeo then had dinner with his senior aides. The next session is due to start at 9am. It is unclear whether Pompeo will meet Kim Jong-un on this trip.

There were reports before Pompeo began his visit, that he might relax the US demand for complete, verifiable, irreversible disarmament (CVID), and settle for mutual confidence-building measures that defused tensions without dismantling the North Korean arsenal.

His spokeswoman, Heather Nauert, denied those reports on Thursday, saying: “Nothing could be further from the truth. Our policy toward North Korea has not changed.

“We are committed to a denuclearised North Korea and Secretary Pompeo looks forward to continuing his consultations with North Korean leaders to follow up on the commitments made at the Singapore summit,” Nauert added.

However, she did explain why Pompeo and other US officials had stopped using the CVID formulation in their remarks.

Pompeo will also use his trip to consult and reassure Washington’s allies in the region, with meetings planned with Japanese and South Korean officials in Tokyo on Sunday. Japan has voiced support for the leaders’ Singapore declaration, but reacted cautiously to Trump’s decision to cancel joint military exercises.